Dynamics of the order parameter statistics in the long range Ising model

Dear Editor,

We would like to thank all the referees for taking their valuable time to review our manuscript and provide constructive comments. Following the comments from the referees we have made significant changes in our manuscript, most notably two appendices (appendix A and appendix B) that compare our numerical results with analytical and exact numerical results at two extremes of our model ($\alpha = \infty$ and $\alpha = 0$ respectively). These changes suggested by the referees have refined our manuscript and we believe it now deserves a publication in SciPost Physics. Below, we address the questions and comments raised by all referees :

include analytical data at $\alpha = \infty$ and compare with numerics (from Report 1)

Response: To address this we have added Appendix A. The Probability Distribution Function (PDF) for nearest neighbor transverse field Ising chain ($\alpha = \infty$ limit in long range Ising model) has been analytically calculated by means of a relation to the partition function of a 3-state classical model. In this appendix we compare the PDF of subsystem magnetization, after quantum quench, obtained numerically with the analytical expression of PDF at stationary state for the same set of quench parameters. Furthermore, we also compare the numerically obtained evolution of formation probabilities ($P(m = \mp \frac{l}{2})$ and $P(m = 0)$) with the analytical results. The comparison shows an extremely good overlap between our TDVP results with the analytical expression, as far as $\alpha$ is getting sufficiently large, as expected.

if possible : include analytical data at $\alpha = 0$ and compare with numerics (from Report 1)

Response: To address this we have added Appendix B. The long range Ising model reduces to a fully connected Ising model at $\alpha = 0$. This model can be represented as the model of a single collective spin. Our initial state is a $\mathbb{Z}_2$ symmetric GHZ state which is in maximal total spin sector. Furthermore, the time evolution operator is unitary and commutes with the Hamiltonian so, through out the time evolution the evolved state remains in the maximal total spin sector. Owing to these properties we exactly diagonalize the Hamiltonian in the sector with maximum total spin. Following this we represent our time evolved state as the superposition of the basis states with time dependent coefficients. Finally, to compute the generating function, we represent each basis states uniquely as the sum of the tensor product between the subsystem part and its complement which is an eigenket of subsystem magnetization operator. We then calculate the PDF from the generating function by a Fourier transformation. The PDF obtained this way is numerically exact. In this appendix we compare our TDVP results with the exact results for two different quench parameters and obtained a high degree of overlap for both PDF and formation probabilities.

comment on the low bond dimensions used (from Report 1)

Response: To address this we have added a brief discussion in the third paragraph of subsection 2.3 and further discussion in Appendix D. We plot the relative error of subsystem magnetization at three increasing values of bond dimension ($\chi = 100$ being the largest) and observe that the error is less than $O(10^{-1})$ overall and less than $O(10^{-3})$ for smaller time scales where most of our results are based. Furthermore, we have also added a plot that shows the absolute error in emptiness formation probability and observe that the error is less than $O(10^{-2})$.

discussion on experimental realization (from Report 3)

Response: To address this we have added a brief discussion about the experimental realization of our model and calculation of the PDF of the order parameter in Subsection 2.4. We start by citing important out of equilibrium ion trap experiments on long range Ising model. We mention that by using the single shot spin detection method we can directly measure the subsystem magnetization at a given time slice after a quantum quench and by repeating the experiment for a statistically significant number of times and collecting the histogram we can measure the PDF. We end the section by citing the experimental works where similar statistical distribution of different parameters are measured.

additional information from order parameter statistics (from Report 3)

Response: To address this we have added a paragraph at the end of subsection 2.2. Here, we briefly discuss the relevance of full counting statistics to study different phenomenon like, electron transport, entanglement, criticality, and many body localization.

We have made the following specific changes in our manuscript:

1. Added last paragraph in subsection 2.2 .
2. Added third paragraph in subsection 2.3 .
3. Added subsection 2.4 .
4. Added Appendix A and Figure 8 .
5. Added Appendix B and Figure 9 .
6. Added Figure 12 in Appendix D .
7. Cited the articles numbered 43,44,45,46,47,48,57,58,59,60,69 in bibliography section.

Please refer to Authors comments section for the detailed explanation on each of these changes.